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JEE Main20231 Feb 2023Evening ShiftMathematicsInverse Trigonometric FunctionsActual

Let S = x ∈ R : 0 < x < 1 and 2 tan - 1 1 - x 1 + x = cos - 1 1 - x 2 1 + x 2 . If n ( S ) denotes the number of elements in S then :

Options

  1. An ( S ) = 2 and only one element in S is less then 1 2
  2. Bn ( S ) = 1 and the element in S is more than 1 2
  3. Cn ( S ) = 1 and the element in S is less then 1 2
  4. Dn ( S ) = 0

Correct answer

C. n ( S ) = 1 and the element in S is less then 1 2

Step-by-step solution

Given, 0 < x < 1 2 tan - 1 1 - x 1 + x = cos - 1 1 - x 2 1 + x 2 Put x = tan θ         ;   0 < θ < π 4 2 tan - 1 1 - tan   θ 1 + tan   θ = cos - 1 1 - tan 2 θ 1 + tan 2 θ ⇒ 2 tan - 1 tan π 4 - θ = cos - 1 ( cos 2 θ ) 2 π 4 - θ = 2 θ ⇒ θ = π 8 Put θ = tan - 1 x we get, x = tan π 8 ∴ x = 2 - 1 ≃ 0 . 414 < 1 2

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